Talk:Prime number/Draft

What kind of number

The first sentence doesn't specify what kind of numbers we are dealing with. Andres Luure 22:26, 5 November 2007 (CST)

It says "A prime number is a number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself." The word "itself" implies it must be a positive whole number. (But maybe it could be more explicit.) Michael Hardy 14:03, 20 December 2007 (CST)

And why not be more explicit? It seems much clearer to me, a number theorist even, if you say "a prime number is a positive whole number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself". In the original version, I didn't immediately return to the beginning of the sentence and think, "oh, that IMPLIES that the original number was a positive whole number". If the only concern is that repeating the same phrase twice might be a turn off, let me quote from Strunk and White's "Elements of Style", one of the guides to style that we are supposed to take advice from (see section 19) "The likeness of form enables the reader to recognize more readily the likeneses of content and function...the unskilled writer often violates this principle mistakenly believing in the value of constantly varying the form of expression. When repeating a statement to emphasize it, the writer may need to vary its form. Otherwise, the writer should follow the principle of parallel construction." It seems to me that the repetition in our case is not for emphasis, and after the repetition, it will be firmly fixed in the reader's mind that all numbers begin considered are positive whole numbers.Barry R. Smith 20:40, 29 March 2008 (CDT)

I am not sure it is even clear to say "exactly two" when it could be "two and only two." --Thomas Simmons 19:56, 8 November 2007 (CST)

In mathematics, at least, the phrases "there are exactly two" and "there are precisely two" are understood to express the same statement as "there are two, and only two" (for instance, see the discrete math text I taught out of this past term, or the Wiki page on if and only if). I have considered these as equivalent for many years, so it is hard for me to put myself in the shoes of someone who might be seeing this for the first time. The issue that you are concerned with is that someone might accidentally confuse "exactly two" with the idea that it has at least two, but possibly more, positive divisors. I cannot see how even people with very little mathematical experience would interpret "exactly two" in this manner, the word "exactly" being inserted exactly (hehe) to let you know that this is the precise number. Furthermore, I think it is hard to argue against "exactly two" being the more elegant phrase. I much prefer the phrasing of the first sentence of the approved article to the first sentence of the current draft. Any rebuttals?Barry R. Smith 20:40, 29 March 2008 (CDT)

I agree. Even to naive readers "exactly two" cannot possibly mean "at least two". J. Noel Chiappa 22:12, 29 March 2008 (CDT)

Footnotes versus links to stubs: Until many of the terms are explained with their own articles, the use of footnotes to explain terms and analogies should continue. Otherwise we will have dead links in red letters for a long time to come. This will also mean that the article can reach a broader population as it is written. The chemistry metaphor is another example, that comparisons might be lost on anyone who is not up to speed on freshman level chemistry. So for ease of use and market appeal and just plain educational focus, explaining terms in footnotes would be a good idea. The high school students using CZ today will be the grad students referring to it in future.--Thomas Simmons 19:56, 8 November 2007 (CST)

What i miss

There are some things, that are not in thearticle:

• Prime numbers and Pseudoprimes (Fermat pseudoprime, Euler pseudoprime, Carmichael number, ...)
• Properties of Prime numbers
  • p is a Prime number <=> p|(p over n) for 1<n<p
  • Prime numbers and Perrin sequence
  • Prime numbers aund Lucas sequence

--arbol01 05:04, 1 January 2008 (CST)

I don't understand the first comment under "Properties of prime numbers"

As for prime divisors of elements of those two particular sequences, it seems to me that these are far too specialized to be included in this page, and would be better placed on the "Perrin sequence" and "Lucas sequence" pages separately. Otherwise, one would need to enumerate ALL named recursively defined sequences, and the divisibility properties in each case. I would imagine that just this task would encompass many pages in itself.Barry R. Smith 00:20, 30 March 2008 (CDT)

1 revisited

Regarding my above comment in "What kind of number" above, I personally feel that the first sentence should read something like, "A prime number is a whole number greater than 1 that can be evenly divided by exactly two positive whole numbers, namely 1 and itself". It seems that the main argument above against saying a prime must be bigger than 1 from the outset is the need for clarity in the first sentence, but I feel that currently this clarity comes at the price of correctness. As written, I feel the that first sentence is plain wrong, and I personally wouldn't put my stamp of approval on it.

I don't want to sound TOO dismissive. I didn't just go edit the draft, because I understand that their was some discussion about this above. Apparently, the status of 1 seems to have been problematic even when the fundamental of arithmetic were laid down in Euclid's "Elements". However, it seems to me that the tone of the approved version suggests that the typical modern "choice" to label 1 as neither prime nor composite is a result of whimsy or chance. This is a false impression.

In a sense, I guess, defining 1 as special can seem as arbitrary as defining 0 factorial to be 1. But with the invention of the gamma function and the recognition of its canonical properties, can there be any dispute as to the correct definition of 0 factorial? Similarly, there are very sound reasons that 1 has been given special status over the last century or so. The easiest to explain is that the Fundamental Theorem of Arithmetic is just false if 1 is considered prime: considere, 6 = 2*3 = 2*3*1 -- two different prime factorizations. (By the way, I also think that the words "Fundamental Theorem of Arithmetic" should appear somewhere on the "prime number" page -- can't remember if I saw it anywhere). A second reason is that with the development of algebraic number theory, the units in algebraic number fields were found to play a very special and important role. Within the integers, 1 and -1 are the only units, so it is hard to get a feel for the special role they play only within this context. Nevertheless, the fact that 1 is the unique multiplicative identity within the integers should make a strong impression. (For more about 1, see this website http://mathforum.org/kb/message.jspa?messageID=1379707, and especially the comments by John Conway, a world-renown number theorist.)

In summary, although the status of 1 might have fluctuated in the past, I believe the consensus of the vast majority of working mathematicians at present is that it should not be considered prime, and this is reflected in todays high-school textbooks. Furthermore, I do not see any indication that this will change soon. Thus, it seems that the proper definition should make it clear that 1 is not prime from the first sentence. Otherwise, we will be spreading disinformation to those casual learners who wonder, "hmm, I wonder if 1 is a prime", look at the first line of the Citizendium page, and then wander off to tell their friends what they learned.Barry R. Smith 01:31, 30 March 2008 (CDT)

Dude, you're the expert! I (at least, can't speak for everyone) defer to your clear familiarity. So I'd go for it. Plus to which, your point about the Fundamental Theorem of Arithmetic is good (and so easily understandably by all that it should probably be mentioned in the article as a reason why 1 is not considered by mathematicians as being part of the set of prime numbers, even though by the simplistic definition of 'prime', it seems to be prime). J. Noel Chiappa 11:40, 30 March 2008 (CDT)

Barry, the first sentence currently says "A prime number is a positive whole number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself." I believe this does say that 1 is not prime, just as you want, as the number 1 has only one divisor, namely 1 itself. So I'm not sure what your point is.

I agree with all the rest you wrote (for what it's worth, as you know of course more number theory than I do). -- Jitse Niesen 16:00, 30 March 2008 (CDT)

Someone careful and analytic might draw that conclusion, but not all our readers might fit that definition. Baldly saying '1 is not a prime number' is probably what they need. Without in any way intended to be demeaning to them, I am always mindful of that wonderful George Carlin line: "Think of how dumb the average person is - and then realize that half of them are dumber than that."

(Adding "different" - as in "two different positive whole numbers" - might make the definition cast-iron, though). But it might still be useful to have a section on 'why 1 is not a prime number'; the point about the Fundamental Theorem of Arithmetic could go there. J. Noel Chiappa 23:31, 30 March 2008 (CDT)

Yes, Jitse, it seems that after all of that, I understand 1 okay but I still have trouble counting to 2 :-). Anyway, my own error emphasizes the point that inferring information about the prime number from information presented at the END of the sentence is not my own thought process, and probably not a lot of other people's. (I suppose if I still kept up my German, I would be used to that sort of thing :-) ). For instance, the end of the sentence in the approved version is where you find that the prime in question is a positive whole number, but I prefer the draft version where it comes right out and tells you that. I think a similar modification to clarify that the whole number is bigger than 1 from the outset, "baldly" saying it, as Noel suggested, is also in order. I also like your suggestion, Noel, of providing clearer reasons for 1's unique position. Would that be better as a new subsection, a footnote, or a link to a page about the arithmetic properties of 1?Barry R. Smith 22:31, 31 March 2008 (CDT)

I'd say a new subsection, not a footnote. Although I don't know where it would fit... hmmmmm (cogitates). Maybe take the third para of the intro, about factorization, and move it to a new section immediately after the intro, titled something like "Factorization and primes"; I think that's a sufficiently important aspect of primes that it's worth of a section on its own. Mention of the Fundamental Theorem of Arithmetic would go there, after which it would be natural to flow from that into your point about the FToA ruling out 1 as a prime. The existing text about "(although this is a matter of [the] definition [of a prime], and mathematicians in the past often did consider 1 to be a prime)" would naturally fit in there too. In fact, maybe a sub-section of that "Factorization and primes" section would cover the primality of 1, and although it would start with the FToA point, etc, you could add your other points above about algebraic number theory, etc. J. Noel Chiappa 00:38, 1 April 2008 (CDT)

Yes, counting is hard ;) I added "greater than 1" to the first sentence, so that's settled for now.

Noel's suggestion to have a new section on the Fundamental Theorem of Arithmetic looks like a good idea. We probably don't want to write too much on it, I think details should go at unique prime factorization or some other article, but I agree that it's important enough in this context to get a section. Indeed, the primality of 1 can covered there, though I'm not sure it should be a sub-section; how much should we say about it? -- Jitse Niesen 08:10, 1 April 2008 (CDT)

Since I'm not a mathematician, and the article is intended (mostly!) for non-mathematicians, would you like me to try the layout I suggested; you all can then check it to make sure I didn't commit any math howlers? J. Noel Chiappa 10:32, 1 April 2008 (CDT)

Sounds good to me Noel Barry R. Smith 11:40, 1 April 2008 (CDT)

OK, I've taken a crack at it. I hope you will all find the result (mostly :-) satisfactory; it seems to me (at least :-) to flow well, and in a natural progression. A couple of things where I don't have enough math knowledge to really fill in, and you all need to backstop: i) explain some about why and how the FToA is so important, ii) some of the more advanced stuff about why 1 is not a prime (in Barry's original comments in this section above) was way over my head, so I just cut-n-pasted the brief allusion here, which you all ought to expand a teensy bit (and make sure my copyediting didn't produce bogosities). Oh, also, the section on factorization should include a sentence or two about how factorization of very large numbers is a key in the crypto-system stuff we alluded to in the intro. I'm too lazy to do that - off to other things! J. Noel Chiappa 12:53, 1 April 2008 (CDT)

I think it looks great, Noel. The only concern I have is the statement that the Fundamental Theorem of Arithmetic is an important building block in many areas of number theory. Historically, the Fundamental Theorem appeared in Euclid's "Elements", the most influential math book of all time, as Proposition 14 in Book IX (This is from a secondary source). Actually, this proposition only shows that if a number n factors as n = p_1 x p_2 x p_3 x ... x p_r, where p_1, ..., p_r are DISTINCT prime numbers (i.e., n could be 30 = 2 x 3 x 5, but not 12 = 2 x 2 x 3, since 2 appears twice), then then those are the only prime numbers that appear in its factorization. Thus, this says significantly less than the Fundamental Theorem of Arithmetic, and only says something about very special types of numbers.

It wasn't until about 2000 years after Euclid that the Fundamental Theorem was codified and decisively proved, by Carl Friedrich Gauss (I have seen this claim many times, but don't have a math historian to use as a source). It seems generally believed that earlier people understood the principle of unique factorization, but perhaps there had never been a reason to try to prove it. It wasn't until larger number systems than the integers began to be considered that it was realized that the Fundamental Theorem describes a particular property of the integers. In fact, in other number systems, the analog of unique factorization FAILS to be true, which is what Gauss realized and motivated him to prove the theorem for integers. So in a sense, it is the failure of the Fundamental Theorem to be an important result in these other number systems (i.e., it's just not true) that prompted its formulation.

Does this make sense? If so, then maybe I will just stick a brief mention of some of this information in place of the statement that I objected to. In any case, besides being an assumed property of the integers that is used to build up many of the important results in Arithmetic, I suppose an answer to your question of why FToA is important is that it fails in other number systems. In response to your other question, I don't see any "bogosities" :). ...said Barry R. Smith (talk) 17:42, 1 April 2008 (Please sign your talk page posts by simply adding four tildes, ~~~~.)

Got it. My text about the Fundamental Theorem of Arithmetic, which is a key building block in many important areas of number theory was in large part a reaction to the very name - I figured anything called the Fundamental Theorem of Arithmetic had to be important! But I notice you say "besides being an assumed property of the integers that is used to build up many of the important results in Arithmetic", so perhaps I wasn't so far wrong? :-)

So, I'll change the text to say "Fundamental Theorem of Arithmetic, which is used to build up many of the important results in the area of arithmetic", and you can further tweak that to your satisfaction, to be perfectly accurate.

After thinking about it, I would suggest that this article probably isn't the place to mention how the FToA is not true in other number systems, because it's one further step removed from the article's focus, which is primes. It would also intrude into the flow from i) the mention of FToA to ii) how the FToA makes it desirable to exclude 1 from the set of primes. That observation would of course be a perfect fit in the Fundamental Theorem of Arithmetic article, though.

I'll also add that remark about how factorization is what's important in public-key crypto work. And then I leave it to you all... :-) J. Noel Chiappa 19:28, 1 April 2008 (CDT)

I do have some remarks and questions.

1. You added the word "different" to the first sentence, so that it reads: "A prime number is a whole number greater than 1 that can be evenly divided by only two different positive whole numbers, namely 1 and itself." Is this necessary? Is there really a chance that if we remove the word "different", somebody will think that 7 is not a prime because it has three divisors, namely 1, 7 and 1?
2. We have an article about unique factorization. Do we need a different article about the Fundamental Theorem of Arithmetic?
3. You say that the Fundamental Theorem of Arithmetic "is used to build up many of the important results in the area of arithmetic." I would replace "arithmetic" by "number theory". Arithmetic can mean number theory, but I think this meaning is disappearing. Barry, would do you think?
4. Finally, I have my doubts about "a more general trend in mathematics over the past century, which is to recognize that 0 and 1 are very special numbers". I thought mathematicians always recognized this. I think the point in the posts you refer to is that mathematicians are getting more careful to make sure that their proofs are really correct.

I'd write longer paragraphs, but that's personal I guess. I have a maths history book which says that Gauss proved the Fundamental Theorem. I think earlier proofs exist, but they are nowadays deemed incomplete. I didn't know that the Elements contains a more restricted result. -- Jitse Niesen 14:18, 3 April 2008 (CDT)

Reponses:

1. I agree, the word different is superfluous.
2. No, we don't need two separate articles. However, if the main article is going to be "unique factorization", then "FToA" should be given as a synonym in the first sentence. Furthermore, typing "FToA" into CZ's main search box should deposit one on that page. Right now, if one searches for "FToA", he just gets a list of search hits, and the first one is actually the prime number page draft, not the unique factorization page. It needs to be redirected -- is this easy? I'll look into it... (Interestingly, typing "unique factorization" into the Wikipedia search box deposits you on a page about unique factorization domains -- a bad choice of redirection IMO).
3. Noel originally wrote "number theory", and my long winded response boiled down to my thinking "arithmetic" was the more apt word. Perhaps it should say "elementary number theory". The problem is that number theory these days is big, and for instance, it seems hard to me to draw a direct connection between unique factorization and major results in analytic number theory (although Euler's factorization of the zeta function involves it). Also, it is precisely the failure of unique factorization that spurred the invention of rings and ideals and algebraic number theory in general (although unique factorization into prime IDEALS is an important building block in this area). Certainly, if you stick to elementary number theory, working with integers and congruences, then it is important, although even here it is hard to gauge how much. Very few proofs seem to come out and say, "and this next step follows from unique factorization". It is more "fundamental" in its importance, since much structure would be absent if it were false. For instance, statements that "such and such type of number has a prime factor of this type" would be silly, if you could have different prime factorizations. Also, security of certain cryptosystems is based on the assumption that the product of two large primes is hard to factor. If there were other factorizations, some that maybe involved small primes, these would no longer be viable cryptosystems. There must be a pithy way to summarize this type of fundamental importance accurately, and if someone thinks of one, that would be great. Maybe a nice allusion to that atoms/molecules metaphor again? I'll try to figure one out myself.
4. I agree that even mathematical noobs probably have always realized that 0 and 1 are "special" numbers right away. However, even important number theorists as late as the 1900's would sometimes list 1 as a prime number. No one found a real need for a proof of FToA until Gauss, so I wouldn't say necessarily that the problem was that proofs were incorrect. I would guess that once Gauss proved this result, he did not consider 1 to be prime (I'd hope so, at least). Perhaps a lot of the time, people didn't realize the appropriate definitions until enough of the theory had been developed, like the general theory of rings and the FToA in this case ("appropriate" definitions being ones that make the theory and theorems as simple and elegant as possible).

I get worried about repeating something like "Gauss was the first to prove this theorem". I have been tempted to write sentences like this several times now, and even if I see one in a history book, there never seems to be a source. How could there be? Someone would have had to comb every extant reference to make sure no one else had proved it earlier, an impossible task. So every statement like this seems to beg the question, "how many old primary sources did you consult before you decided that Gauss was the first?" It certainly seems to happen regularly that some new source is discovered that proves such a statement false.Barry R. Smith 22:04, 3 April 2008 (CDT)

A few addtional bits:

1. I added that "different" after thinking for a while of how to make it as clear and consise as possible for non-mathematician readers. Yes, technically it's superfluous (to the likes of us :-) - but I believe it will increase the likelihood of correct comprehension for the 'average' reader.
2. For what it's worth, Wikipedia has a separate article on the Fundamental Theory of Mathematics. Not saying we need one, that's y'all's call, just providing data. I have set up the redirects as you suggest.
3. I believe I can find a way to say exactly what Barry wants - that although it's not called out specifically, what it says is important, and the basic idea/attribute of the integers (unique factorizability) is used throughout "elementary number theory" (which I will also put in).
4. That was my (perhaps not very good) attemtp to capture the essence of what Conway said in one of his posts in that thread ("Mathematicians this century are generally much more careful about exceptional behavior of numbers like 0 and 1 than were their predecessors: we nowadays take care to adjust our statements so that our theorems are actually true. It's easy to find lots of statements in 19th century books that are actually false with the definitions their authors used"). I tried to capture the jist of what he said in a concise way that would make sense to non-experts, although perhaps I didn't succeed - I will look at it again.

Thanks for being so patient with this non-mathematician! :-) J. Noel Chiappa 23:02, 3 April 2008 (CDT)

Primes of special forms subsection

The third type of prime considered in this section seems out of place to me. As far as I know, primes of the form n^2+1 are mostly a curiosity, and uninteresting for anything else. It would be easy to find many other types of "primes" of this form. Any one object to me removing them? As consolation, I am going to insert a bullet about primes in arithmetic sequences, which seem much more important to me.Barry R. Smith 17:50, 1 April 2008 (CDT)

I might suggest leaving them because Mersenne primes are a major source of "largest prime" values. I know, I know, to most true mathematicians this is probably boring juvenile crap, but some people enjoy it (like some people enjoy tiddliwinks) - see, for instance, GIMP so an article on primes ought to at least mention Mersenne primes! Blast, wrote that whole section, then realized you were talking about N^2+1, not 2^N-1! On thinking about it, they seem rather obscure to me, so we can probably lose them. I'd just comment them out, rather than discard the text completely; that way, if someone makes a case for them, it's trivial to put them back. J. Noel Chiappa 19:40, 1 April 2008 (CDT)

Alternative definition

I am interested in having a non-mathematician perspective on the last part of this section. It seems to me to ramp up in sophistication very quickly, starting with mention of the word "ideal", and then moving into sentences about "rings" and "generation" of "ideals". Certainly, if this page is intended for non-specialists, then those terms should at least be linked. But would they be better placed in a page about prime ideals in rings, and a much simplified discussion put in its place on this page?Barry R. Smith 17:50, 1 April 2008 (CDT)

I couldn't make heads or tails of it most of it. I would just move everything past the Euclid's Lemma thing to an /Advanced subpage (see Quantum mechanics/Advanced for what such pages look like). The first couple of paras I would leave, as they are comprehensible to us mortals, and mildly interesting - but I will rewrite them a bit to make them clearer. J. Noel Chiappa 19:48, 1 April 2008 (CDT)

Fermat primes

The current discussion of Fermat primes says that one can construct a regular p-gon if p is a Fermat prime. Perhaps more surprising is that these are the ONLY primes for which you can construct a regular p-gon, so I am going to add this.

Other types of primes

I think that if we are going to discuss other types of primes, obvious choices include Wieferich and Wilson primes. But where do we draw a line about which special types of primes to include? Wieferich primes showed up in work on Fermat's Last Theorem. Perhaps a criterion would be to include any special forms for which a significant result is known? I like this better than the criterion of including any forms of primes with "names".Barry R. Smith 18:15, 1 April 2008 (CDT)

Write articles on them, and link to those articles from the "Related Articles" subpage. As you suggest, I'd only put really important ones here. That's probably where to put Fermat primes, now that I think of it. I'll leave that to you all too. J. Noel Chiappa 19:48, 1 April 2008 (CDT)